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1 There are three kinds of people in the world those who are good at math and those who are not. PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 Positive Views The record of a month s roulette playing at Monte Carlo can afford us material for discussing the foundations of knowledge. --Karl Pearson Not-so-positive views I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive. --Plato There are three kinds of lies: lies, damned lies, and statistics. --Benjamin Disraeli (British Prime Minister) popularized by Mark Twain Statistics: a set of mathematical procedures for summarizing and interpreting observations Allows us to find order in apparent chaos Data: observations made about the environment Numbers that we wish to understand and summarize Descriptive Statistics: a branch of statistics used to summarize or describe a set of observations a branch of statistics used to interpret or draw inferences about a set of observations 1

2 Why does measurement matter in statistics? Statistics deal only with numbers, and thus, we must understand what these numbers actually represent Statistics Data (numbers/measurements) Knowledge (inferences made from data about underlying processes using probability) Measurement: the application of any rule by which numbers are assigned to cases in order to represent the presence/absence or quantity of some attribute possessed by each case Nominal: a set of mutually exclusive categories according to their characteristics -each is assigned an arbitrary numerical code which conveys qualitative information -IDENTITY -ex. gender, species, political party, religious affiliation Ordinal: reflects the ordering of cases on the attribute being measured -gives a rough indication of qualitative differences -equal score differences DO NOT necessarily reflect equal differences in the amount of the attribute being measured -IDENTITY & ORDER -ex. finishing positions in a race (e.g., 1st & 2nd) & class rank 2

3 Interval: equal score differences DO reflect equal differences in the amount of the attribute being measured -scores reflect the number of fixed-sized units of the attribute possessed by each case -positioning of the zero-point is entirely arbitrary a score of zero does NOT mean the case possesses none of the attribute which means it is NOT possible to say how many times higher one score is than another -IDENTITY, ORDER, & EQUAL DISTANCE -ex. IQ, SAT score, Celsius scale & Fahrenheit scale Ratio: possesses all the characteristics of interval scale but has a non-arbitrary zeropoint. Thus, a case scored as zero on a ratio scale possesses none of the attribute being measured -IDENTITY, ORDER, EQUAL DISTANCE, & TRUE ZERO POINT -ex. age, volume, Kelvin temperature scale Many of the most commonly used statistical methods (e.g., t-test, ANOVA, correlation, regression) assume that data are measured at least on an interval scale Some nonparametric methods can be used for data measured on an ordinal scale or even a nominal scale (e.g., chi-square test) 3

4 Central Tendency: refers to the "middle" value or most typical value Measures include the mean, median, and mode Dispersion (or Variability): describes the spread or clustering of a set of data Measures include the variance, standard deviation, and range Rectangular Distribution Normal Distribution Bimodal Distribution 4

5 Population Target group for inference Examples (all women, all 4 th graders, & everyone who has experienced emotional maltreatment) Parameter: numerical characteristic of population. Example: population mean is µ (mu). Sample Subgroup of the population Examples (women in this class, 4 th graders at a local elementary school, & USM students who report emotional maltreatment during mass testing) Statistic: numerical characteristic of sample. Example: sample mean is (X-bar). POPULATION: All of the individuals of interest The results from the sample are generalized to the population The sample is selected from the population SAMPLE: The individuals selected to participate in the research study Sampling: Process of selecting the sample from the population Random sampling: Independent selection As contrasted to evenness Descriptive vs. Inferential Statistics Descriptive: primary purpose is to describe some aspect of the data Inferential: primary purpose is to infer (to estimate or to make a decision, test a hypothesis) Sample Infer Population Statistic Infer Parameter 5

6 Population Parameters Sample Statistics Mean µ Variance σ 2 s 2, s* 2 Standard Deviation σ s, s* Correlation ρ r All inferential statistics have the following in common: Use of some descriptive statistic Use of probability Potential for estimation Sampling variability Sampling distributions Use of a theoretical distribution Two hypotheses, two decisions, & two types of error Use of Some Descriptive Statistic Video Delay of gratification (Walter Mischel) Results: EI 517s, EN 365s, OI 585s, ON 590s The sample mean, a descriptive statistic, has value 365 for group EN Coping Ideas Ideas None Rewards Exposed Obscured

7 Use of Probability Coin toss example Fair coin? p(head)=.5 What if I toss the coin 10 times and get 10 heads? p(10 H fair coin)=1/1024 = Reject hypothesis of fair coin What if I toss the coin 10 times and get 6 heads? p(6 or more H fair coin) =386/1024=.377 Retain hypothesis of fair coin Potential for Estimation The sample mean ( ) is a descriptive statistic that has the value 365 for group EN and it is an estimate of the population mean (µ) which is a population parameter for population EN. Coping Ideas Ideas None Rewards Exposed Obscured Sampling Variability and Sampling Distributions If we ran the delay of gratification study again, we probably would not get exactly 365 for the mean of the EN group Sampling Distribution of The value of the new mean might be close, like 355 A third study might yield 375, etc. Sampling distribution is a distribution of statistics rather than scores 1,000 Samples 7

8 Use of a Theoretical Distribution Normal Distribution Two Hypotheses, Two Decisions, and Two Types of Error Two hypotheses Coin is fair OR coin is not fair Kids in the EN group score lower OR they do not Null Hypothesis (H 0 ) Alternative Hypothesis (H 1 ) Two decisions Reject or retain the null hypothesis These decisions always concern the null hypothesis Two types of error Type I error: reject H 0 when H 0 is true Type II error: retain H 0 when H 0 if false (H 1 is true) All inferential statistics have the following in common: Use of some descriptive statistic Use of probability Potential for estimation Sampling variability Sampling distributions Use of a theoretical distribution Two hypotheses, two decisions, & two types of error 8

9 Null hypothesis significance testing begins with researchers assuming that their predictions are wrong Simple two-group experimental design would assume no real difference between control group and experimental group The null hypothesis is being tested Statistical significance focuses on the probability that a statistic of a particular magnitude would emerge by chance alone A statistically significant result is one that is extreme enough that it is unlikely to have occurred by chance alone A number of factors can influence whether results are statistically significant Greater effect size Higher alpha (α) level (willingness to mistakenly reject H 0 when H 0 is true [Type I error]) The more willing you are to risk making a Type I error, then the more likely you are to find statistical significance Decrease measurement error (e.g., use within-subject designs) Greater sample size Avoid restriction of range Use of directional hypotheses These have greater power than non-directional hypotheses if you are correct in predicting the direction of the effect Effect Size: an estimate of the magnitude or meaningfulness of an effect For example, Cohen s d is a measure of effect size that captures how different two means are in terms of their standard deviation Meta-Analysis: a set of statistical techniques that are used to summarize and evaluate entire sets of research These approaches analyze the results of studies rather than the responses of participants 9